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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $\\mathfrak g$ be a Lie algebra over $k$.\nLet $(x_i)_{i \\in I}$ be a [[basis|0KCQ9Y3QKA]] of $\\mathfrak g$ indexed by a totally ordered set $I$.\nLet $U(\\mathfrak g)$ be the [[universal enveloping algebra|SCZ8PRVY00]] of $\\mathfrak g$ with its canonical map $\\iota \\colon \\mathfrak g \\to U(\\mathfrak g)$.\nThen the ordered monomials\n$$\\iota(x_{i_1}) \\iota(x_{i_2}) \\cdots \\iota(x_{i_m}), \\qquad m \\ge 0, \\quad i_1 \\le i_2 \\le \\cdots \\le i_m,$$\nform a basis of $U(\\mathfrak g)$ over $k$, where the monomial with $m = 0$ is $1$.\n\n## Remarks\n\nThat the ordered monomials span is easy: any product of the $\\iota(x_i)$ can be reordered.\nEach swap of neighbours changes it by a product of lower length, by the relation $xy - yx = [x, y]$.\nLinear independence is the hard part.\nA standard proof builds an action of $\\mathfrak g$ on the polynomial ring $S(\\mathfrak g)$ that sends each ordered monomial applied to $1$ to the matching polynomial monomial.\nThe Jacobi identity is used to check that this is an action.\nIn particular $\\iota$ is injective, so every Lie algebra over a field is a Lie subalgebra of an associative algebra with the commutator bracket.\nEquivalently, the [[associated graded algebra|8HFAEGFQT1]] of $U(\\mathfrak g)$ for its [[filtration|R1PTHQACWM]] by length of products is isomorphic to the [[symmetric algebra|PBGF9TXMB7]] $S(\\mathfrak g)$.\nExample: for $\\mathfrak{sl}_2$ with basis $f \u003c h \u003c e$, the elements $f^a h^b e^c$ with $a, b, c \\ge 0$ form a basis of $U(\\mathfrak{sl}_2)$.\nOver a commutative ring instead of a field, the theorem holds when $\\mathfrak g$ is free as a module, but it can fail in general.\n\n## Explanations\n\n### L1 human\n\nThe enveloping algebra is defined by generators and relations, and such definitions can collapse.\nThe theorem says that nothing collapses here: the enveloping algebra is exactly as large as a polynomial ring in a basis of the Lie algebra.\nYou may write every element as a combination of monomials with the basis elements in a fixed order, and this way of writing it is unique.\n\n### L2 human\n\nThe theorem is the basic structural fact about $U(\\mathfrak g)$.\nIt shows that $\\mathfrak g$ embeds in $U(\\mathfrak g)$, that $U(\\mathfrak g)$ has no zero divisors, and that $U(\\mathfrak g)$ is Noetherian when $\\mathfrak g$ is finite-dimensional, by passing to the polynomial ring $S(\\mathfrak g)$.\nFor a subalgebra $\\mathfrak h \\subseteq \\mathfrak g$, it shows that $U(\\mathfrak g)$ is a free $U(\\mathfrak h)$-module, which is the key input for Verma modules and induced representations.\nIt also gives the vector space isomorphism $S(\\mathfrak g) \\to U(\\mathfrak g)$ by symmetrisation in characteristic $0$.\n",
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